Additive Group of Integers

Theorem

The set of integers under addition $\left({\Z, +}\right)$ forms a countably infinite abelian group.

Thus it follows that integer addition is:

• Well-defined on $\Z$
• Closed on $\Z$
• Associative on $\Z$
• Commutative on $\Z$
• The identity of $\left({\Z, +}\right)$ is $0$
• Each element of $\left({\Z, +}\right)$ has an inverse.

Proof

Proof of Abelian Group

From the definition of the integers, the algebraic structure $\left({\Z, +}\right)$ is an isomorphic copy of the inverse completion of $\left ({\N, +}\right)$.

As the Natural Numbers are a Naturally Ordered Semigroup, it follows that:

• $\left ({\N, +}\right)$ is a commutative semigroup;
• all elements of $\left ({\N, +}\right)$ are cancellable.

The result follows from Inverse Completion of Commutative Semigroup is Abelian Group.

Thus addition on $\Z$ is well-defined, closed, associative and commutative on $\Z$.

$\blacksquare$

Let us define $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ as in the formal definition of integers.

That is, $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$ is an equivalence class of ordered pairs of natural numbers under the congruence relation $\boxminus$.

$\boxminus$ is the congruence relation defined on $\N \times \N$ by:

$\left({x_1, y_1}\right) \boxminus \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$

In order to streamline the notation, we will use $\left[\!\left[{a, b}\right]\!\right]$ to mean $\left[\!\left[{\left({a, b}\right)}\right]\!\right]_\boxminus$, as suggested.

Identity is Zero

From Construction of Inverse Completion: Identity of Quotient Structure, the identity of $\left({\Z, +}\right)$ is $\left[\!\left[{c, c}\right]\!\right]$ for any $c \in \N$:

	\(\displaystyle \)	\(\displaystyle \forall a, b, c \in \N:\)	\(\displaystyle \)	\(\displaystyle \)	\(\)	\(\displaystyle \left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{c, c}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{a, b}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{c, c}\right]\!\right] + \left[\!\left[{a, b}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

$\left[\!\left[{c, c}\right]\!\right]$ is the equivalence class of pairs of elements $\N \times \N$ whose difference is zero.

Thus the identity of $\left({\Z, +}\right)$ is seen to be $0$.

Note that a perfectly good representative of $\left[\!\left[{c, c}\right]\!\right]$ is $\left[\!\left[{0, 0}\right]\!\right]$. This usually keeps to a minimum the complexity of any arithmetic that is needed.

$\blacksquare$

Construction of Inverses

From Construction of Inverse Completion: Invertible Elements in Quotient Structure, we see that every element of $\left({\Z, +}\right)$ has an inverse.

We can see that:

	\(\displaystyle \)	\(\displaystyle \forall a, b \in \N:\)	\(\displaystyle \)	\(\displaystyle \left[\!\left[{a, b}\right]\!\right] + \left[\!\left[{b, a}\right]\!\right]\)	\(=\)	\(\displaystyle \left[\!\left[{a + b, b + a}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left[\!\left[{a + b, a + b}\right]\!\right]\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Natural Number Addition is Commutative

The above construction is valid because $a$ and $b$ are both in $\N$ and hence cancellable.

From Construction of Inverse Completion: Identity of Quotient Structure, $\left[\!\left[{a + b, a + b}\right]\!\right]$ is a member of the equivalence class which is the identity of $\left({\Z, +}\right)$.

Thus the inverse of $\left[\!\left[{a, b}\right]\!\right]$ is $\left[\!\left[{b, a}\right]\!\right]$.

$\blacksquare$

Integers are Countably Infinite

Finally we note that from Integers are Countable, the set of integers can be placed in one-to-one correspondence with the set of natural numbers.

$\blacksquare$

Sources

• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.5$: Example $80$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$: Example $7.1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.3$: Example $1$
• Ian D. Macdonald: The Theory of Groups (1968): $\S 1$: Example $1.01$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 32$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 34 \ (1)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.4$